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Theorem iooval2 13353

Description: Value of the open interval function. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

**Assertion**
Ref	Expression
iooval2	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

**Proof of Theorem iooval2**
Step	Hyp	Ref	Expression
1		iooval 13344	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ^ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
2		elioore 13350	. . . . . 6 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
3	2	ssriv 3978	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ ℝ
4	1, 3	eqsstrrdi 4029	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → {𝑥 ∈ ℝ^ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ⊆ ℝ)
5		df-ss 3957	. . . 4 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ⊆ ℝ ↔ ({𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ∩ ℝ) = {𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
6	4, 5	sylib 217	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ({𝑥 ∈ ℝ^ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ∩ ℝ) = {𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
7		inrab2 4299	. . . 4 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ∩ ℝ) = {𝑥 ∈ (ℝ^* ∩ ℝ) ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)}
8		ressxr 11254	. . . . . 6 ⊢ ℝ ⊆ ℝ^*
9		sseqin2 4207	. . . . . 6 ⊢ (ℝ ⊆ ℝ^* ↔ (ℝ^* ∩ ℝ) = ℝ)
10	8, 9	mpbi 229	. . . . 5 ⊢ (ℝ^* ∩ ℝ) = ℝ
11	10	rabeqi 3437	. . . 4 ⊢ {𝑥 ∈ (ℝ^* ∩ ℝ) ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)}
12	7, 11	eqtri 2752	. . 3 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ∩ ℝ) = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)}
13	6, 12	eqtr3di 2779	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → {𝑥 ∈ ℝ^ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
14	1, 13	eqtrd 2764	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3424 ∩ cin 3939 ⊆ wss 3940 class class class wbr 5138 (class class class)co 7401 ℝcr 11104 ℝ^*cxr 11243 < clt 11244 (,)cioo 13320

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-pre-lttri 11179 ax-pre-lttrn 11180

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-ioo 13324

This theorem is referenced by: elioo2 13361 ioomax 13395 ioopos 13397 dfioo2 13423

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